On the Fine-Grained Complexity of Empirical Risk Minimization: Kernel Methods and Neural Networks
This provides foundational insights into the inherent computational limits of ERM, which is crucial for algorithm designers and theorists in machine learning.
The paper tackles the computational complexity of empirical risk minimization (ERM) for kernel methods and neural networks, showing under complexity-theoretic assumptions that no sub-quadratic time algorithms exist for solving these problems to high accuracy.
Empirical risk minimization (ERM) is ubiquitous in machine learning and underlies most supervised learning methods. While there has been a large body of work on algorithms for various ERM problems, the exact computational complexity of ERM is still not understood. We address this issue for multiple popular ERM problems including kernel SVMs, kernel ridge regression, and training the final layer of a neural network. In particular, we give conditional hardness results for these problems based on complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis. Under these assumptions, we show that there are no algorithms that solve the aforementioned ERM problems to high accuracy in sub-quadratic time. We also give similar hardness results for computing the gradient of the empirical loss, which is the main computational burden in many non-convex learning tasks.